15. Law of Cosines

Solving an SAS Triangle The Law of Sines was good for ASA - two angles and the included side AAS - two angles and any side SSA - two sides and an opposite angle (being aware of possible ambiguity) Why would the Law of Sines not work for an SAS triangle? 15 No side opposite from any angle to get the ratio 26° 12.5

SAS Use the law of cosines formula to find the side across from the angle you know Then use the law of sines to find the smaller angle of the two that are left Then subtract 2 angles you know from 180 to find the last angle

Deriving the Law of Cosines Triangle ABC does not have a right angle so we drop a perpendicular from Angle B (same as yesterday) We call this h and now we have a right angle so we can use SOHCAHTOA B c a h r A C b We also call the section on the bottom left r Using SOHCAHTOA, sinA=h/c so h=c sinA cosA=r/c so r=c cosA Now use the Pythagorean theorem for triangle on the right (see next slide)

Deriving the law of cosines This same process could be used for the other sides to come up with their law of cosines

Law of Cosines

Example (SAS)– solve the triangle C 15 26° 12.5 A B c

Example continued Now calculate the angles You must do the smallest one first C 15 26° 12.5 A B c = 6.65

SSS If you are given all 3 sides, you must find the angle across from the largest side first!!!! After that, you can use the law of sines to find another angle. Then subtract the two angles you just found from 180 to find the last angle.

ExAMPLE (SSS) Example Solve triangle ABC if a = 9.47 feet, b =15.9 feet, and c = 21.1 feet.

Example cont. Use the law of sines or the law of cosines To show that C  70.1°. Then,

Summary of Cases with Suggested Procedures Case 1:SAA or ASA Suggested Procedure for Solving Find the remaining angle using the angle sum formula (A + B + C = 180°). Find the remaining sides using the law of sines. Case 2: SSA Suggested Procedure for Solving This is the ambiguous case; 0, 1, or 2 triangles. Find an angle using the law of sines. Find the remaining angle using the angle sum formula. Find the remaining side using the law of sines. If two triangles exist, repeat steps 2 and 3.

Summary of Cases with Suggested Procedures CASE 3: SAS Suggested Procedure for Solving Find the third side using the law of cosines. Find the smaller of the two remaining angles using the law of sines. Find the remaining angle using the angle sum formula. CASE 4: SSS Suggested Procedure for Solving Find the largest angle using the law of cosines. Find either remaining angle using the law of sines. Find the remaining angle using the angle sum formula.

FINDING AREA FOR LAW OF COSINES The law of cosines can be used to derive a formula for the area of a triangle given the lengths of three sides known as Heron’s Formula. If a triangle has sides of lengths a, b, and c and if the semiperimeter is Then the area of the triangle is

Using Heron’s Formula to Find an Area Example The distance “as the crow flies” from Los Angeles to New York is 2451 miles, from New York to Montreal is 331 miles, and from Montreal to Los Angeles is 2427 miles. What is the area of the triangular region having these three cities as vertices?

Cost of a Lot An industrial piece of real estate is priced at $4.15 per square foot. Find, to the nearest $1000, the cost of a triangular lot measuring 324 feet by 516 feet by 412 feet. 324 412 516